Let $G$ be a connected reductive $\mathbb{Q}$-group. Let $\mathbb{A}$ denote the ring of adèles of $\mathbb{Q}$. Let $B \subset G(\mathbb{A})$ be a compact, let $x \in G(\mathbb{A})$ and consider the counting problem \begin{equation*} \sum_{\gamma \in G(\mathbb{Q})} \mathbf{1}_{B}(x^{-1} \gamma x) = \text{card} \big( G(\mathbb{Q}) \cap xBx^{-1} \big) \end{equation*} where $\mathbf{1}_B$ is the indicator function of $B$. Let $U \subset G(\mathbb{A})$ be a symmetric neighborhood of the identity. Following [Gorodnik, Nevo - Counting lattice points, Lemma 2.1], we have \begin{equation*} \text{card} \big( G(\mathbb{Q}) \cap xBx^{-1} \big) \leq \int_{B^x_+} \phi_U (g^{-1}) dg \end{equation*} where $B^x_+ = U (xBx^{-1}) U$ and \begin{equation*} \phi_U (g) = \sum_{\gamma \in G(\mathbb{Q})} \text{vol}(U)^{-1} \mathbf{1}_U (\gamma g) \quad (g \in G(\mathbb{A})). \end{equation*}
In the above, we fixed a Haar measure on $G(\mathbb{A})$ as follows: let $P \subset G$ be a minimal $\mathbb{Q}$-parabolic subgroup, with Langlands decomposition $P = NMA$ such that $G(\mathbb{A}) = P (\mathbb{A}) K$, where $K = \prod_p K_p K_{\infty}$ (with open compact subgroups $K_p \subset G(\mathbb{Q}_p)$ and maximal compact subgroup $K_{\infty} \subset G(\mathbb{R})$). Let $A(\mathbb{R})^0$ denote the identity component of the $\mathbb{R}$-points of $A$, then any $x \in G(\mathbb{A})$ can be writen as \begin{equation*} x = nmak \quad n \in N(\mathbb{A}), m \in M(\mathbb{A}), a \in A(\mathbb{R})^0, k \in K. \end{equation*} Let $dn, dm, da$ and $dk$ denote Haar measures on $N(\mathbb{A})$, $M(\mathbb{A})$, $A(\mathbb{R})^0$ and $K$ respectively, there is a Haar measure $dg$ on $G(\mathbb{A})$ such that \begin{equation*} \int_{G(\mathbb{A})} f(g) dg = \int_{N(\mathbb{A})} \int_{M(\mathbb{A})} \int_{A(\mathbb{R})^0} \int_{K} a^{2 \rho} f(nmak) dkdadmdn \quad (f \in C_c(G(\mathbb{A}))) \end{equation*} where $\rho$ denote the half-sum of positive roots.
Let $\omega \subset N(\mathbb{A}) M(\mathbb{A})$ be compact subset and let $x \in \omega A(\mathbb{R})^0 K$.
Question: Is it true that, writing $x = nmak$, there is a constant $c > 0$ (independent of $x$) such that \begin{equation*} \text{vol} (B^x_+) \leq c \text{vol}(B) a^{2 \rho} \end{equation*} holds ? In particular \begin{equation*} \text{card} \big( G(\mathbb{Q}) \cap xBx^{-1} \big) \leq c \text{vol}(B) a^{2 \rho} \end{equation*} for some $c > 0$ independent of $x$ ?
